ANU BDSI
workshop Generalised Linear Models
Biological Data Science Institute
27th September 2024
Genralised linear models (GLMs) has three components:
| Link | \eta_i = g(\mu_i) | \mu_i = g^{-1}(\eta_i) |
|---|---|---|
| Identity | \mu_i | \eta_i |
| Log | \log(\mu_i) | \exp(\eta_i) |
| Inverse | 1/\mu_i | 1/\eta_i |
| Inverse-squared | 1/\mu_i^2 | 1/\sqrt{\eta_i} |
| Square-root | \sqrt{\mu_i} | \eta_i^2 |
| Logit | \log(\mu_i/(1-\mu_i)) | \exp(\eta_i)/(1+\exp(\eta_i)) |
| Probit | \Phi^{-1}(\mu_i) | \Phi(\eta_i) |
| Complementary log-log | \log(-\log(1-\mu_i)) | 1-\exp(-\exp(\eta_i)) |
where \Phi(\cdot) is the cumulative distribution function of the standard normal distribution.
| Family | Canonical Link | Range of Y_i | V(Y_i|\eta_i) |
|---|---|---|---|
| Normal | Identity | (-\infty, \infty) | \psi |
| Binomial | Logit | [0, 1] | \mu_i(1-\mu_i) |
| Poisson | Log | [0, \infty) | \mu_i |
| Gamma | Inverse | (0, \infty) | \psi\mu_i^2 |
| Inverse-Gaussian | Inverse-squared | (0, \infty) | \psi\mu_i^3 |
where \psi is the dispersion parameter, \eta_i is the linear predictor, and \mu_i is the expectation of Y_i. In the binomial family, n_i is the number of trials.
anu-bdsi.github.io/workshop-GLM/